I'm going to push the time frame back into the first century BC if that's OK. The survival of 900+ of Cicero's letters mean that we have a very good idea of how letter writing and sending worked in the last few decades of the Republic, and to be honest it's not likely that very much changed into the early imperial period (apart from the more formal official system introduced by Augustus, but which you can't use for private letters anyway).

As you've moved to Egypt, you are likely to be writing on papyrus (made from thin strips of the plant, which were formed into sheets and glued together into rolls: instructions here) using ink made from cuttlefish ink, soot or other such substances. You're likely to be writing with a pen made from reed. Elsewhere in the Empire, as well as in Egypt, a huge variety of other writing materials were used. Wax tablets (i.e. pieces of wood covered in thin layers of beeswax into which you scratch your letter), parchment made from animal hide, pot sherds (called ostraka), sheets of wood (like the Vindolanda tablets) and so on.

For delivery, you have three main options: • Your own family, friends, clients, or slaves. • Someone connected to the addressee. For example, if your friend send you a letter first, you could send yours back with whomever delivered the first message. • A third party who happens to be travelling in the right direction.

The status of the letter-carrier will affect the delivery. If you want your letter to get to the recipient quickly then it's probably best to send your own slave, who will ideally do this task quickly and return as soon as possible. Of course this only works if you have a large enough household, otherwise can you afford to spare someone who might be gone for weeks, perhaps even months? Free people have to be given more leeway - depending on their connection to you and their own travel plans, delivering your letter might not be their top priority. On the one hand, they might go somewhere else first, might move slowly or stop off en route. On the other hand, they might leave quickly, before you've properly finished writing or when you don't really have anything to say. It would be considered rude not to send some form of greeting if you've got friends at the destination these people are travelling to, especially so if they send letters to you.

Ideally you would seal your letter, so that the recipient knows it hasn't been tampered with on the way. Again, the status of the courier will determine how much you trust them to deliver the letter without peaking at the contents - your own family members and slaves are perhaps more trustworthy than third parties who happen to be going in the right direction. Cicero regularly apologises to his addressees for not discussing certain topics, usually because he couldn't find trustworthy people to carry them. Here's a very good example from a letter to his best friend Atticus:

"In these letters, indeed, I am urgently pressed by you to send answers, but what renders me rather dilatory in this respect is the difficulty of finding a trustworthy carrier. How few of these gentry are able to convey a letter rather weightier than usual without lightening it by skimming its contents! Besides, I do not always care to send whenever anyone is starting for Epirus: for I suppose that, having offered victims before your Amaltheia, you at once started for the siege of Sicyon. And yet I am not even certain when you start to visit Antonius or how much time you are devoting to Epirus. Accordingly, I don’t venture to trust either Achaeans or Epirotes with a letter somewhat more outspoken than usual. Now some events have occurred since you left me worth my writing to you, but they must not be trusted to the risk of a letter being lost, opened, or intercepted."

Here's another from Cicero to his brother Quintus, who at the time (54 BC) was serving with Julius Caesar in Gaul:

"I would also advise you not to trust anything to a letter which might embarrass us if it became public property. There are many things of which I had sooner be ignorant than informed, if the information carries risk… On your part, please see to it that I know to whom I ought to give the letter which I shall then be sending – should it be Caesar’s couriers, so that he forwards it to you straight away, or Labienus’? For where those Nervii of yours live and how far they are away from us I don’t know. It gave me much pleasure to learn from your letter of the courage and dignity of Caesar’s bearing in his great sorrow. As for your command that I finish the poem addressed to him which I had begun, my time is much distracted and my mind far more so, but since Caesar knows from my letter to you that I have started on something, I shall go back to what I began and finish it during these leisure days of Supplications."

From both of these you can see some of the difficulties of arranging delivery - in the first, Cicero is uncertain about Atticus' exact whereabouts, and in the second he asks his brother for clarification on which couriers to use. Another interesting detail from the second is the fact that there are clearly details of Cicero's letters that he expects his brother to share more widely. Quintus has shown Caesar one of Cicero's letters, in which he tells him that he is writing a poem about Caesar. There are other such examples, where the boundary between private and public are blurred and Cicero is fully expecting his letter to be read by people other than the intended recipient.

In terms of the exact method of delivery, as the whole system was privately arranged it seems that letters were taken directly to the home of the addressee, or at least to wherever they were expected to be.